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Συμμετρική Ομάδα
Συμμετρική Ομάς Symmetry Group thumb|300px| [[Αλγεβρική Ομάδα Ομαδοθεωρία ]] thumb|300px|"Θεαματική" Απεικόνιση της Ομάδας E8 thumb|300px| [[Συμμετρία Τρίγωνο ]] - Μία Αλγεβρική Ομάδα. Ετυμολογία Η ονομασία "ομάδα" σχετίζεται ετυμολογικά με την λέξη "ομού". Εισαγωγή In abstract algebra, the symmetric group S''n'' on a finite set of n'' symbols is the group whose elements are all the permutation operations that can be performed on ''n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself. Low degree groups The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately. ;S0 and S1: The symmetric groups on the empty set and the singleton set are trivial, which corresponds to 1=0! = 1! = 1. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S0, its only member is the Empty function. ;S2: This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and so abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting fs(x'',''y) = f(x'',''y) + f(y'',''x), and fa(x'',''y) = f''(''x,y'') − ''f(y'',''x), one gets that 2·''f'' = f''s + ''f''a. This process is known as symmetrization. ;S3: S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S3 to S2 corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A3 kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents. ;S4: The Symmetric group S4 is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, Rencontres numbers 9, 8 and 6 permutations, of the cube.Die Untergruppenverbände der Gruppen der ordnung weniger als 100, Habilitationsschrift, J. Neubuser, Universität Kiel, Germany, 1967. Beyond the group A4, S4 has a Klein four-group V as a proper normal subgroup, namely the even transpositions {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, with quotient S''3. In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from S4 to S3 also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree n'' of dimension below ''n − 1, which only occurs for 1=''n'' = 4. ;S5: S5 is the first non-solvable symmetric group. Along with the special linear group SL(2, 5) and the icosahedral group A5 × S2, S5 is one of the three non-solvable groups of order 120, up to isomorphism. S5 is the Galois group of the general quintic equation, and the fact that S5 is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map S5 → S6 as a transitive subgroup; the obvious inclusion map S''n'' → S''n''+1 fixes a point and thus is not transitive. This yields the outer automorphism of S6, discussed below, and corresponds to the resolvent sextic of a quintic. ;S6: Unlike all other symmetric groups, S6, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map S5 → S6 as a transitive subgroup (the obvious inclusion map Sn → Sn+1 fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S6—see automorphisms of the symmetric and alternating groups for details. :Note that while A6 and A7 have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of S6 and S7, these do not correspond to exceptional Schur multipliers of the symmetric group. Μητραϊκή Αναπαράσταση της S3 \begin{array}{rcccll} e &=& \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}&=& (1)&\qquad \mathrm{ord}\left(e\right)=1\\ \\ d &=& \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}&=&(1~2~3)&\qquad \mathrm{ord}\left(d\right)=3\\ \\ d^2 &=& \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}&=&(1~3~2)&\qquad \mathrm{ord}\left(d^2\right)=3\\ \\ s_1 &=& \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}&=&(2~3)&\qquad \mathrm{ord}\left(s_1\right)=2\\ \\ s_2 &=& \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}&=&(1~3)&\qquad \mathrm{ord}\left(s_2\right)=2\\ \\ s_3 &=& \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}&=&(1~2)&\qquad \mathrm{ord}\left(s_3\right)=2 \end{array} : GL(2,\mathbb{F}_2) =\left\{ \begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 &1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 &0 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 &1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 &1 \\ 1 & 1 \end{bmatrix} \right\} is isomorphic with S_3 . Υποσημειώσεις Εσωτερική Αρθρογραφία * Ομαδοθεωρία * Μαθηματική Μήτρα * Προσθετική Ομάδα * Πολλαπλασιαστική Ομάδα * Συμμετρική Ομάδα S4 * Συμμετρική Ομάδα S3 * Συμμετρική Ομάδα S2 * Εναλλάσσουσα Ομάδα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *physics.upatras.gr *opinionator.blogs.nytimes.com *"Introduction to Group Theory for Physicists", Marina von Steinkirch Κατηγορία:Αλγεβρικές Ομάδες